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Search: id:A103905
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| A103905 |
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Square array T(n,k) read by antidiagonals: number of tilings of an <n,k,n> hexagon. |
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11
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| 1, 1, 2, 1, 6, 3, 1, 20, 20, 4, 1, 70, 175, 50, 5, 1, 252, 1764, 980, 105, 6, 1, 924, 19404, 24696, 4116, 196, 7, 1, 3432, 226512, 731808, 232848, 14112, 336, 8, 1, 12870, 2760615, 24293412, 16818516, 1646568, 41580, 540, 9, 1, 48620, 34763300
(list; table; graph; listen)
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OFFSET
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1,3
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COMMENT
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As a square array, T(n,k) = number of all k-watermelons without a wall of length n. - S. R. Finch (Steven.Finch(AT)inria.fr), Mar 30 2008
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REFERENCES
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A. J. Guttmann, A. L. Owczarek and X. G. Viennot, Vicious walkers and Young tableaux. I. Without walls, J. Phys. A 31 (1998) 8123-8135.
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LINKS
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P. J. Forrester and A. Gamburd, Counting formulas associated with some random matrix averages
H. Helfgott and I. M. Gessel, Enumeration of tilings of diamonds and hexagons with defects
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FORMULA
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T(n, k) = [V(2n+k-1)V(k-1)V(n-1)^2]/[V(2n-1)V(n+k-1)^2], with V(n) the superfactorial numbers (A000178).
T(n, k) = Prod[j=0..k-1, j!(j+2n)!/(j+n)!^2 ].
T(n, k) = Prod[h=1..n, Prod[i=1..k, Prod[j=1..n, (h+i+j-1)/(h+i+j-2) ]]].
T(n,k)=Prod[i=1..k, Prod[j=n+1..2n+1, i+j]/Prod[j=0..n, i+j]]; - Paul Barry (pbarry(AT)wit.ie), Jun 13 2006
Conjectural formula as a sum of squares of Vandermonde determinants: T(n,k) = 1/((1!*2! ... *(n-1)!)^2*n!)* sum {1 <= x_1, ..., x_n <= k} (det V(x_1, ...,x_n))^2, where V(x_1, ...,x_n} is the Vandermonde matrix of order n. Compare with A133112. - Peter Bala (pbala(AT)toucansurf.com), Sep 18 2007
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EXAMPLE
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Array begins:
1,2,3,4,5,6,
1,6,20,50,105,196,
1,20,175,980,4116,14112,
1,70,1764,24696,232848,1646568,
1,252,19404,731808,16818516,267227532,
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CROSSREFS
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Rows include A002415, A047819, A047835, A047831. Columns include A000984 and A000891. Main diagonal is A008793.
Cf. A133112.
Sequence in context: A128741 A060539 A163269 this_sequence A103209 A089900 A138533
Adjacent sequences: A103902 A103903 A103904 this_sequence A103906 A103907 A103908
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KEYWORD
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nonn,tabl
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AUTHOR
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Ralf Stephan, Feb 22 2005
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